NA1.01illustrate equivalent ratios, using a variety of tools
(e.g., concrete materials, diagrams, dynamic geometry software) (e.g., show that 4:6 represents the same ratio as 2:3 by showing that a ramp with a height of 4 m and a base of 6 m and a ramp with a height of 2 m and a base of 3 m are equally steep);
NA1.02represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations
(Sample problem:You are building a skateboard ramp whose ratio of height to base must be 2:3.Write a proportion that could be used to determine the base if the height is 4.5 m.);
NA1.03solve for the unknown value in a proportion, using a variety of methods
(e.g., if 500 mL of juice costs $2.29, the unit rate is 0.458¢/mL; this unit rate is less than for 750 mL of juice at $3.59, which has a unit rate of 0.479¢/mL);
NA1.05solve problems involving ratios, rates, and directly proportional relationships in various contexts
(e.g., currency conversions, scale drawings, measurement), using a variety of methods (e.g., using algebraic reasoning, equivalent ratios, a constant of proportionality; using dynamic geometry software to construct and measure scale drawings) (Sample problem: Simple interest is directly proportional to the amount invested. If Luis invests $84 for one year and earns $1.26 in interest, how much would he earn in interest if he invested $235 for one year?);
NA1.06solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms
(e.g., calculating simple interest and sales tax; analysing data) (Sample problem: Of the 29 students in a Grade 9 math class, 13 are taking science this semester. If this class is representative of all the Grade 9 students in the school, estimate and calculate the percent of the 236 Grade 9 students who are taking science this semester. Estimate and calculate the number of Grade 9 students this percent represents.).
Overall Expectations:
NAV.02simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations
.
Specific Expectations:
NA2.01simplify numerical expressions involving integers and rational numbers, with and without the use of technology
*The knowledge and skills described in this expectation are to be introduced as needed and applied and consolidated throughout the course;
NA2.02relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations;
NA2.03 describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three
[i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x<sup>2</sup>; volume, which is three dimensional, can be represented by (x)(x)(x), (x<sup>2</sup>)(x), or x<sup>3</sup>];
NA2.04 substitute into and evaluate algebraic expressions involving exponents
(i.e., evaluate expressions involving natural-number exponents with rational-number bases) [e.g., evaluate (3/2)<sup>3</sup> by hand and 9.83 by using a calculator]) (Sample problem: A movie theatre wants to compare the volumes of popcorn in two containers, a cube with edge length 8.1 cm and a cylinder with radius 4.5 cm and height 8.0 cm. Which container holds more popcorn?);* *The knowledge and skills described in this expectation are to be introduced as needed and applied and consolidated throughout the course;
NA2.05add and subtract polynomials involving the same variable up to degree three
[e.g., (2x + 1) + (x<sup>2</sup> - 3x + 4)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
NA2.06multiply a polynomial by a monomial involving the same variable to give results up to degree three
[e.g., (2x)(3x), 2x(x + 3)], using a variety of tools (e.g., algebra tiles, drawings, computer algebra systems, paper and pencil);
NA2.07solve first-degree equations with nonfractional coefficients, using a variety of tools
(e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies) (Sample problem: Solve 2x + 7 = 6x - 1 using the balance analogy.);
NA2.08substitute into algebraic equations and solve for one variable in the first degree
(e.g., in relationships, in measurement) (Sample problem: The perimeter of a rectangle can be represented as P = 2l + 2w. If the perimeter of a rectangle is 59 cm and the width is 12 cm, determine the length.).
Number Sense and Algebra
Overall Expectations :
NAV.01 solve problems involving proportional reasoning;
Specific Expectations:
NA1.01 illustrate equivalent ratios, using a variety of tools
(e.g., concrete materials, diagrams, dynamic geometry software) (e.g., show that 4:6 represents the same ratio as 2:3 by showing that a ramp with a height of 4 m and a base of 6 m and a ramp with a height of 2 m and a base of 3 m are equally steep);NA1.02 represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations
(Sample problem:You are building a skateboard ramp whose ratio of height to base must be 2:3.Write a proportion that could be used to determine the base if the height is 4.5 m.);NA1.03 solve for the unknown value in a proportion, using a variety of methods
(e.g., concrete materials, algebraic reasoning, equivalent ratios, constant of proportionality) (Sample problem: Solve x/4 = 15/20.);NA1.04 make comparisons using unit rates
(e.g., if 500 mL of juice costs $2.29, the unit rate is 0.458¢/mL; this unit rate is less than for 750 mL of juice at $3.59, which has a unit rate of 0.479¢/mL);NA1.05 solve problems involving ratios, rates, and directly proportional relationships in various contexts
(e.g., currency conversions, scale drawings, measurement), using a variety of methods (e.g., using algebraic reasoning, equivalent ratios, a constant of proportionality; using dynamic geometry software to construct and measure scale drawings) (Sample problem: Simple interest is directly proportional to the amount invested. If Luis invests $84 for one year and earns $1.26 in interest, how much would he earn in interest if he invested $235 for one year?);NA1.06 solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms
(e.g., calculating simple interest and sales tax; analysing data) (Sample problem: Of the 29 students in a Grade 9 math class, 13 are taking science this semester. If this class is representative of all the Grade 9 students in the school, estimate and calculate the percent of the 236 Grade 9 students who are taking science this semester. Estimate and calculate the number of Grade 9 students this percent represents.).Overall Expectations:
NAV.02 simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations
.Specific Expectations:
NA2.01 simplify numerical expressions involving integers and rational numbers, with and without the use of technology
NA2.02 relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations;
NA2.03 describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three
[i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x<sup>2</sup>; volume, which is three dimensional, can be represented by (x)(x)(x), (x<sup>2</sup>)(x), or x<sup>3</sup>];NA2.04 substitute into and evaluate algebraic expressions involving exponents
(i.e., evaluate expressions involving natural-number exponents with rational-number bases) [e.g., evaluate (3/2)<sup>3</sup> by hand and 9.83 by using a calculator]) (Sample problem: A movie theatre wants to compare the volumes of popcorn in two containers, a cube with edge length 8.1 cm and a cylinder with radius 4.5 cm and height 8.0 cm. Which container holds more popcorn?);* *The knowledge and skills described in this expectation are to be introduced as needed and applied and consolidated throughout the course;NA2.05 add and subtract polynomials involving the same variable up to degree three
[e.g., (2x + 1) + (x<sup>2</sup> - 3x + 4)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);NA2.06 multiply a polynomial by a monomial involving the same variable to give results up to degree three
[e.g., (2x)(3x), 2x(x + 3)], using a variety of tools (e.g., algebra tiles, drawings, computer algebra systems, paper and pencil);NA2.07 solve first-degree equations with nonfractional coefficients, using a variety of tools
(e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies) (Sample problem: Solve 2x + 7 = 6x - 1 using the balance analogy.);NA2.08 substitute into algebraic equations and solve for one variable in the first degree
(e.g., in relationships, in measurement) (Sample problem: The perimeter of a rectangle can be represented as P = 2l + 2w. If the perimeter of a rectangle is 59 cm and the width is 12 cm, determine the length.).